Three-dimensional isometric tensor networks

Abstract

Tensor network states are expected to be good representations of a large class of interesting quantum many-body wave functions. In higher dimensions, their utility is however severely limited by the difficulty of contracting the tensor network, an operation needed to calculate quantum expectation values. Here we introduce a method for the time-evolution of three-dimensional isometric tensor networks which respects the isometric structure and therefore renders contraction simple through a special canonical form. Our method involves a tetrahedral site-splitting which allows to move the orthogonality center of an embedded tree tensor network in a simple cubic lattice to any position. Using imaginary time-evolution to find an isometric tensor network representation of the ground state of the 3D transverse field Ising model across the entire phase diagram, we perform a systematic benchmark study of this method in comparison with exact Lanczos and quantum Monte Carlo results. We show that the obtained energy matches the exact groundstate result accurately deep in the ferromagnetic and polarized phases, while the regime close to the critical point requires larger bond dimensions. This behavior is in close analogy with the two-dimensional case, which we also discuss for comparison.

arXiv comment15 pages, 17 figures

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• Hilbert curve vs Hilbert space: exploiting fractal 2D covering to increase tensor network efficiency
• Adaptive-weighted tree tensor networks for disordered quantum many-body systems
• Typical Correlation Length of Sequentially Generated Tensor Network States
• Isometric tensor network representations of two-dimensional thermal states
• A Comprehensive Cross-Model Framework for Benchmarking the Performance of Quantum Hamiltonian Simulations
• Exploring Bosonic and Fermionic Link Models on $(3+1)-$d tubes
• Single-layer tensor network approach for three-dimensional quantum systems
• Two Dimensional Isometric Tensor Networks on an Infinite Strip
• Quantum Gauge Networks: A New Kind of Tensor Network
• Optimisation of ultrafast singlet fission in 1D rings towards unit efficiency
• Stacked tree construction for free-fermion projected entangled pair states
• Scalable projected entangled-pair state representation of random quantum circuit states
• Diagonal Isometric Form for Tensor Network States in Two Dimensions
• Pseudoentanglement from tensor networks
• Sampling two-dimensional isometric tensor network states